418 SIR J. F. W. RERSCHEL ON THE ALGEBRAIC EXPRESSION OF THE 
(38.) For Z we have 
9^ 1 
1) 1 
144 ' 144 
It will be convenient to separate this into two parts, viz. 
9(a;-l 
144 
and 
Q'-=rk{- 1 1 • >2.+o. i2,_,+&c.|. 
First, then, for proceeding as in article 30, we have 
%o(^) = -ii4(^-i); Xi(^)=o. 
X(x) =2^.Xo(a:+ 10) + 2^_i%o(^d-5) 
==~l44|(^ + 9).2,+ (^+4).2,_, 
X (^) = — y^(2^H-2^_i) = — 
whence Z' consisting now of the single term R', 
2xd-10j?.2 
x{x+10) 
2880\" 
As regards the other portion of Z, which in this case is R", it has for its expression 
9 y — {\+I^Yx + l'x, ^ l—(l+AVx+9"a; 
_2 - — ^ ^ - J - /'Qj-in n^J_o / 
144 
(9 + 10.0) + 2,_i.-- ‘ (4+10.0). 
Now, whatever be h and c, we have always 
^ (c+ 10.0) = (5 — c)/i— 5/^^ 
In this, if we write for h successively K=p^-\-q^ and K and for c, 9 and 4, we 
find 
- +Ti4{2.(4*'+6A") - 2,_.(A"- 5A'")}. 
But since a?— 10 and s=5, we have 
A'=±|0.10.,+ 1.10^_i + 2.10^_2+3.10^_3 + 4.10^_4-5.10^_5-4.10^_6-3.10^_7-2.10^_8-1.10^_9j 
^' = A|0. 10^+1. 10^-, + 2.10,._2+ + 9.10^_9|. 
Substituting which in Z" and employing the property of equation (14.) for the com- 
